Vaccinate your trees!

For a graph $G$ and an integer-valued function $\tau$ on its vertex set, a dynamic monopoly is a set of vertices of $G$ such that iteratively adding to it vertices $u$ of $G$ that have at least $\tau(u)$ neighbors in it eventually yields the vertex set of $G$. We study two vaccination problems, where the goal is to maximize the minimum order of such a dynamic monopoly either by increasing the threshold value of $b$ vertices beyond their degree, or by removing $b$ vertices from $G$, where $b$ is a given non-negative integer corresponding to a budget. We show how to solve these problems efficiently for trees.